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          算法竞赛准备4----例题（Java语言）
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        <p>几个题目和解析</p>
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<p>废话少说，直接看代码（内部有思路）</p>
<figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br><span class="line">99</span><br><span class="line">100</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> 算法竞赛4 </span>&#123;</span><br><span class="line">    <span class="comment">/*</span></span><br><span class="line"><span class="comment">        1、小白上楼梯。</span></span><br><span class="line"><span class="comment">            共有n阶楼梯，假如每次只能上1、2或3个台阶，问有几种上楼梯的方案。</span></span><br><span class="line"><span class="comment">        问题分析：</span></span><br><span class="line"><span class="comment">            该类问题可以倒着想。</span></span><br><span class="line"><span class="comment">            小白最后一步是在倒数第二、第三、第四个台阶。</span></span><br><span class="line"><span class="comment">            总方案 = 到倒数第二个台阶的方案 + 到倒数第三个台阶的方案 + 到倒数第四个台阶的方案。</span></span><br><span class="line"><span class="comment">            f(n) = f(n - 1 ) + f(n - 2) + f(n - 3)</span></span><br><span class="line"><span class="comment">            得到递推公式，明显，该表达式为一递归表达式。</span></span><br><span class="line"><span class="comment"></span></span><br><span class="line"><span class="comment">            递归的三步走：</span></span><br><span class="line"><span class="comment">                找重复、找变化、找边界</span></span><br><span class="line"><span class="comment">            边界值：当n = 1时，返回1，f(1) = f(0) + f(-1) + f(-2)</span></span><br><span class="line"><span class="comment">                   当n = 2时，返回2，f(2) = f(1) + f(0) + f(-1)</span></span><br><span class="line"><span class="comment">                   当n = 3时，返回4，f(3) = f(2) + f(1) + f(0)</span></span><br><span class="line"><span class="comment">            当n &lt; 0时，返回0；</span></span><br><span class="line"><span class="comment">            当n = 0时，返回1</span></span><br><span class="line"><span class="comment">     */</span></span><br><span class="line">    <span class="function"><span class="keyword">static</span> <span class="keyword">int</span> <span class="title">f1</span><span class="params">(<span class="keyword">int</span> n)</span></span>&#123;</span><br><span class="line">        <span class="keyword">if</span>(n == <span class="number">0</span>)</span><br><span class="line">            <span class="keyword">return</span> <span class="number">1</span>;</span><br><span class="line">        <span class="keyword">else</span> <span class="keyword">if</span>(n &lt; <span class="number">0</span>)</span><br><span class="line">            <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">        <span class="keyword">else</span></span><br><span class="line">            <span class="keyword">return</span> f1(n - <span class="number">1</span>) + f1(n - <span class="number">2</span>) + f1(n - <span class="number">3</span>);</span><br><span class="line">    &#125;</span><br><span class="line">    </span><br><span class="line">    <span class="meta">@Test</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">void</span> <span class="title">testF1</span><span class="params">()</span></span>&#123;</span><br><span class="line">        System.out.println(f1(<span class="number">4</span>));</span><br><span class="line">        </span><br><span class="line">        <span class="comment">//断言，需要外部包。可以忽略。</span></span><br><span class="line">        <span class="comment">//Assertions.assertThat(f1(4)).isEqualTo(7);</span></span><br><span class="line">    &#125;</span><br><span class="line">    </span><br><span class="line">    <span class="comment">/*</span></span><br><span class="line"><span class="comment">        2、旋转数组的最小数字（改造二分法）</span></span><br><span class="line"><span class="comment">            把一个数组最开始的若干个元素搬到末尾,称为数组的旋转</span></span><br><span class="line"><span class="comment">            输入一个递增排序的数组的一个旋转，输出旋转数组的最小元素</span></span><br><span class="line"><span class="comment">            例如&#123;3，4，5，1，2&#125;为&#123;1，2，3，4，5&#125;的旋转数组</span></span><br><span class="line"><span class="comment">            该数组的最小值为1</span></span><br><span class="line"><span class="comment"></span></span><br><span class="line"><span class="comment">            有序：用二分法</span></span><br><span class="line"><span class="comment">            把数组用中间值分成2段，看两端那一段无序（左侧的大于右侧的），最小值在无序的那一段。</span></span><br><span class="line"><span class="comment">            结束时，只剩下两个数，右侧的一定是最小值。</span></span><br><span class="line"><span class="comment">     */</span></span><br><span class="line">    <span class="function"><span class="keyword">static</span> <span class="keyword">int</span> <span class="title">f2</span><span class="params">(<span class="keyword">int</span>[] arr)</span></span>&#123;</span><br><span class="line">        <span class="keyword">int</span> begin = <span class="number">0</span>;</span><br><span class="line">        <span class="keyword">int</span> end = arr.length - <span class="number">1</span>;</span><br><span class="line">        <span class="keyword">int</span> mid = (begin + end) / <span class="number">2</span>;</span><br><span class="line">        <span class="keyword">while</span>(begin + <span class="number">1</span> &lt; end)&#123;</span><br><span class="line">            <span class="keyword">if</span>(arr[begin] &gt; arr[mid])&#123;</span><br><span class="line">                end = mid;</span><br><span class="line">                mid = (begin + mid) / <span class="number">2</span>;</span><br><span class="line">            &#125;</span><br><span class="line">            <span class="keyword">else</span> &#123;</span><br><span class="line">                begin = mid;</span><br><span class="line">                mid = (mid + end) / <span class="number">2</span>;</span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">return</span> arr[end];</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="meta">@Test</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">void</span> <span class="title">testF2</span><span class="params">()</span></span>&#123;</span><br><span class="line">        <span class="keyword">int</span>[] arr = &#123;<span class="number">5</span>,<span class="number">6</span>,<span class="number">7</span>,<span class="number">2</span>,<span class="number">3</span>,<span class="number">4</span>&#125;;</span><br><span class="line">        System.out.println(f2(arr));</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    </span><br><span class="line">    </span><br><span class="line">     <span class="comment">/*</span></span><br><span class="line"><span class="comment">        3、求一个部分有序的数组中，最长的连续递增子序列的长度。</span></span><br><span class="line"><span class="comment">            例如（1，3，4，23，23，5，4，2，3，5，6，7，8）</span></span><br><span class="line"><span class="comment">                 其最长子序列为（2，3，5，6，7，8）</span></span><br><span class="line"><span class="comment">            思路：</span></span><br><span class="line"><span class="comment">                由于要求其最长子序列，及其长度。</span></span><br><span class="line"><span class="comment">                使用双指针对数组进行操作。</span></span><br><span class="line"><span class="comment">     */</span></span><br><span class="line">     <span class="function"><span class="keyword">static</span> <span class="keyword">int</span> <span class="title">len</span><span class="params">(<span class="keyword">int</span>[] arr)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">int</span> first = <span class="number">0</span>, last = <span class="number">0</span>;</span><br><span class="line">        <span class="keyword">int</span> len = <span class="number">0</span>;</span><br><span class="line">        <span class="keyword">while</span> (first &lt; arr.length - <span class="number">1</span>) &#123;</span><br><span class="line">            <span class="keyword">if</span> (arr[first] &gt; arr[first + <span class="number">1</span>]) &#123;</span><br><span class="line">                <span class="keyword">if</span> (first - last + <span class="number">1</span> &gt; len ) &#123;</span><br><span class="line">                    len = first - last + <span class="number">1</span>;</span><br><span class="line">                &#125;</span><br><span class="line">                last = first + <span class="number">1</span>;</span><br><span class="line">                first = first + <span class="number">1</span>;</span><br><span class="line">                <span class="keyword">continue</span>;</span><br><span class="line">            &#125;</span><br><span class="line">            first++;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">if</span>(arr[first] &gt; arr[first - <span class="number">1</span>])&#123;</span><br><span class="line">            <span class="keyword">if</span> (first - last + <span class="number">1</span> &gt; len ) &#123;</span><br><span class="line">                len = first - last + <span class="number">1</span>;</span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">return</span> len;</span><br><span class="line">    &#125;</span><br></pre></td></tr></table></figure>


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